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Error analysis of the space-time DGFEM for nonstationary nonlinear convection-diffusion problems
The paper will be concerned with the numerical solution of nonstationary problems with nonlinear convection as well as diffusion by the space-time discontinuous Galerkin finite element method (DGFEM). The time interval is split into subintervals and on each time level a different space mesh with hanging nodes may be used in general. In the discontinuous Galerkin formulation we use the nonsymmetric, symmetric or incomplete version of the discretization of the diffusion terms and interior and boundary penalty (i.e., NIPG, SIPG or IIPG versions). For the space and time discretization, piecewise polynomial approximations of different degrees p and q, respectively, are used. We assume that the diffusion coefficient depends on the sought solution, but we do not allow its degeneration. The abstract error estimate is derived with the use of the so-called discrete characteristic function. Under the assumption that the triangulations on all time levels are uniformly shape regular, and the exact solution has some regularity properties, error estimates are derived for the space-time DGFEM.